Differentially private federated learning (DP-FL) suffers from slow convergence under tight privacy budgets due to the overwhelming noise introduced to preserve privacy. While adaptive optimizers can accelerate convergence, existing second-order methods such as DP-FedNew require O(d^2) memory at each client to maintain local feature covariance matrices, making them impractical for high-dimensional models. We propose DP-FedSOFIM, a server-side second-order optimization framework that leverages the Fisher Information Matrix (FIM) as a natural gradient preconditioner while requiring only O(d) memory per client. By employing the Sherman-Morrison formula for efficient matrix inversion, DP-FedSOFIM achieves O(d) computational complexity per round while maintaining the convergence benefits of second-order methods. Our analysis proves that the server-side preconditioning preserves (epsilon, delta)-differential privacy through the post-processing theorem. Empirical evaluation on CIFAR-10 demonstrates that DP-FedSOFIM achieves superior test accuracy compared to first-order baselines across multiple privacy regimes.

翻译：差分隐私联邦学习（DP-FL）在严格的隐私预算下，由于为保护隐私而引入的过量噪声，存在收敛速度缓慢的问题。虽然自适应优化器可以加速收敛，但现有的二阶方法（如DP-FedNew）要求每个客户端维护局部特征协方差矩阵，需要O(d^2)的内存开销，这使得它们对于高维模型而言不切实际。我们提出了DP-FedSOFIM，一种服务器端的二阶优化框架，它利用费舍尔信息矩阵（FIM）作为自然梯度预处理器，同时每个客户端仅需O(d)的内存。通过采用Sherman-Morrison公式进行高效的矩阵求逆，DP-FedSOFIM在每轮计算中实现了O(d)的计算复杂度，同时保持了二阶方法的收敛优势。我们的分析证明，通过后处理定理，服务器端的预处理操作保持了(ε, δ)-差分隐私。在CIFAR-10数据集上的实证评估表明，与多种隐私机制下的一阶基线方法相比，DP-FedSOFIM取得了更优的测试精度。